A literature review has shown that there exist adequate techniques to obtain ground reaction curves for tunnels excavated in elastic-brittle and perfectly plastic materials However, for strain-softening materials it seems that the problem has not been sufficiently analysed In this paper, a one-dimensional numerical solution to obtain the ground reaction curve (GRC) for circular tunnels excavated in strain-softening materials is presented



The problem is formulated in a very general form and leads to a system of ordinary differential equations By adequately defining a fictitious ‘time’ variable and re-scaling some variables the problem is converted into an initial value one, which can be solved numerically by a Runge–Kutta–Fehlberg method, which is implemented in MATLAB environment



The method has been developed for various common particular behaviour models including Tresca, Mohr–Coulomb and Hoek–Brown failure criteria, in all cases with non-associative flow rules and two-segment piecewise linear functions related to a principal strain-dependent plastic parameter to model the transition between peak and residual failure criteria



Some particular examples for the different failure criteria have been run, which agree well with closed-form solutions—if existing—or with FDM-based code results Parametric studies and specific charts are created to highlight the influence of different parameters



The proposed methodology intends to be a wider and general numerical basis where standard and newly featured behaviour modes focusing on obtaining GRC for tunnels excavated in strain-softening materials can be implemented This way of solving such problems has proved to be more efficient and less time consuming than using FEM- or FDM-based numerical 2D codes Copyright © 2003 John Wiley & Sons, Ltd

Ground response curves for rock masses exhibiting strain-softening behaviour

Satellite synthetic aperture radar (SAR) interferometry shows that the magnitude 7.6 Manyi earthquake of 8 November 1997 produced a 170-kilometer-long surface break with up to 7 meters of left-lateral slip, reactivating a N76°E quaternary fault in western Tibet. The radar interferometric map reveals asymmetric, along-strike displacement profiles between the two sides of the surface rupture, a pattern that cannot be explained with linear elastic theory. This observation suggests that the elastic moduli of the crust in tension and in compression are different because of the presence of cracks in the crust at shallow depth. A model indicates that a ratio of 2 between compressive and tensile elastic moduli can account for the observed asymmetry, a ratio that is consistent with laboratory and borehole measurements.

https://science.sciencemag.org/content/sci/286/5438/272.full.pdf

Evidence of Nonlinear Elasticity of the Crust from the Mw7.6 Manyi (Tibet) Earthquake

A study of the mechanical behaviour of coal for pillar design

[1] The coseismic deformation due to the 1992 M(w)7.3 Landers earthquake, southern California, is investigated using synthetic aperture radar (SAR) and Global Positioning System (GPS) measurements. The ERS-1 satellite data from the ascending and descending orbits are used to generate contiguous maps of three orthogonal components ( east, north, up) of the coseismic surface displacement field. The coseismic displacement field exhibits symmetries with respect to the rupture plane that are suggestive of a linear relationship between stress and strain in the crust. Interferometric synthetic aperture radar (InSAR) data show small-scale deformation on nearby faults of the Eastern California Shear Zone. Some of these faults ( in particular, the Calico, Rodman, and Pinto Mountain faults) were also subsequently strained by the 1999 M(w)7.1 Hector Mine earthquake. I test the hypothesis that the anomalous fault strain represents essentially an elastic response of kilometer-scale compliant fault zones to stressing by nearby earthquakes [Fialko et al., 2002]. The coseismic stress perturbations due to the Landers earthquake are computed using a slip model derived from inversions of the InSAR and GPS data. Calculations are performed for both homogeneous and transversely isotropic half-space models. The compliant zone model that best explains the deformation on the Calico and Pinto Mountain faults due to the Hector Mine earthquake successfully predicts the coseismic displacements on these faults induced by the Landers earthquake. Deformation on the Calico and Pinto Mountain faults implies about a factor of 2 reduction in the effective shear modulus within the similar to 2 km wide fault zones. The depth extent of the low-rigidity zones is poorly constrained but is likely in excess of a few kilometers. The same type of structure is able to explain high gradients in the radar line of sight displacements observed on other faults adjacent to the Landers rupture. In particular, the Lenwood fault north of the Soggy Lake has likely experienced a few centimeters of left-lateral motion across < 1-km-wide compliant fault zone having the rigidity reduction of more than a factor of 2. The inferred compliant fault zones are interpreted to be a result of extensive damage due to past earthquakes.

/pdf/probing-the-mechanical-properties-of-seismically-active-50ngj2eipz.pdf

Probing the mechanical properties of seismically active crust with space geodesy: Study of the coseismic deformation due to the 1992 Mw7.3 Landers (southern California) earthquake

https://eost.unistra.fr/fileadmin/upload/EOST/Mike_Heap/pdf_files/Heap_and_Faulkner_2008_IJRMMS.pdf

Quantifying the evolution of static elastic properties as crystalline rock approaches failure

Porous or clastic rocks often have elastic moduli which are not constant but increase with increasing minor principal stress. The use of classical constant modulus linear elasticity in these cases can lead to erroneous predictions of the deformations and of the initiation and extent of failure around underground excavations. To illustrate these effects, solutions are developed for axisymmetric excavations in infinite media having power law and exponential variations of elastic modulus with minor principal stress. The maximum stress concentrations do not occur at the excavation boundaries and are less than the constant value of 2.0 given by constant modulus elasticity. When modified slightly to allow for test boundary conditions, the theory gives predictions that are consistent with aspects of the results obtained in hydrostatic compression tests on thick walled cylinders of three sedimentary rocks.

J. W. Bray

Papers

Influence of stress-dependent elastic moduli on stresses and strains around axisymmetric boreholes